TITLE: L_p norm balls for 1<p<2 - is it always similar to an L_q norm ball for some q>2?
QUESTION [8 upvotes]: The L_1 ball in 2D is shaped like a diamond (L_1 is also known as the Manhattan norm). The L_∞ ball is shaped like a square (L_∞ is also known as the supremum norm). They are similar, i.e. have same shape. The L2 ball is shaped like a circle.
Hypothesis: For all p in the interval (1,2), there is q>2 such that the q-ball and the p-ball are similar. One further hypothesis is that this occurs precisely when p,q are Hölder conjugates.
I wasn't sure how to tag this.

REPLY [10 votes]: $\ell_p^n$ is isometric to $\ell_r^n$ with $1\le p < r\le \infty$ iff $n=1$ or ($n=2$ and $p=1$ and $r=\infty$ and the scalars are real). While known before L. Dor's 1976 paper in the Israel J. Math. 24, 260-268, it can be deduced from Theorem 2.1 in that paper, which classifies when $\ell_s^n$ isometrically embeds into $L_t(0,1)$.